Problem: Simplify the following expression: $y = \dfrac{-7x^2+19x+36}{x - 4}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-7)}{(36)} &=& -252 \\ {a} + {b} &=& &=& {19} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-252$ and add them together. Remember, since $-252$ is negative, one of the factors must be negative. The factors that add up to ${19}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-9}$ and ${b}$ is ${28}$ $ \begin{eqnarray} {ab} &=& ({-9})({28}) &=& -252 \\ {a} + {b} &=& {-9} + {28} &=& 19 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({-7}x^2 {-9}x) + ({28}x +{36}) $ Factor out the common factors: $ x(-7x - 9) - 4(-7x - 9)$ Now factor out $(-7x - 9)$ $ (-7x - 9)(x - 4)$ The original expression can therefore be written: $ \dfrac{(-7x - 9)(x - 4)}{x - 4}$ We are dividing by $x - 4$ , so $x - 4 \neq 0$ Therefore, $x \neq 4$ This leaves us with $-7x - 9; x \neq 4$.